\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  -15.0  &   & +  7.00 x_{2} & -3.00 x_{3} & +  4.00 x_{4} & +  6.00 x_{5} & -2.00 x_{6} & +  6.00 x_{7} & +  8.00 x_{8}\\
 x_{10}   &  -3.0 & +  1.00 x_{1} & +  8.00 x_{2} & -9.00 x_{3} & -3.00 x_{4} & +  7.00 x_{5} & +  3.00 x_{6} & +  9.00 x_{7} & + 10.00 x_{8}\\
 x_{11}   &  12.0 & +  2.00 x_{1} & +  5.00 x_{2} & -7.00 x_{3} & +  9.00 x_{4} & +  7.00 x_{5} & +  7.00 x_{6} & +  2.00 x_{7} & -4.00 x_{8}\\
 x_{12}   &  -36.0 & -5.00 x_{1} & +  8.00 x_{2} & +  6.00 x_{3} & -2.00 x_{4} & -6.00 x_{5} & +  3.00 x_{6} & +  5.00 x_{7} & +  9.00 x_{8}\\
 x_{13}   &  64.0 & -8.00 x_{1} & -10.00 x_{2} & -6.00 x_{3} & + 10.00 x_{4} & +  5.00 x_{5} & -1.00 x_{6} & -7.00 x_{7} & -5.00 x_{8}\\
 x_{14}   &  7.0 & +  1.00 x_{1} & +  4.00 x_{2} & +  5.00 x_{3} &   & +  7.00 x_{5} & -3.00 x_{6} & -4.00 x_{7} & -1.00 x_{8}\\
 x_{15}   &  -4.0 & +  6.00 x_{1} & +  3.00 x_{2} & + 10.00 x_{3} & +  3.00 x_{4} & -7.00 x_{5} & -9.00 x_{6} & +  1.00 x_{7} & -2.00 x_{8}\\
\hline
z    &  0.0 & +  2.00 x_{1} & +  2.00 x_{2} & +  3.00 x_{3} & +  1.00 x_{4} & -2.00 x_{5} & +  4.00 x_{6} & -3.00 x_{7} & -3.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0  &   & -1.00 y_{10} & -2.00 y_{11} & +  5.00 y_{12} & +  8.00 y_{13} & -1.00 y_{14} & -6.00 y_{15}\\
 y_{2}   &  1.0 & -7.00 y_{9} & -8.00 y_{10} & -5.00 y_{11} & -8.00 y_{12} & + 10.00 y_{13} & -4.00 y_{14} & -3.00 y_{15}\\
 y_{3}   &  1.0 & +  3.00 y_{9} & +  9.00 y_{10} & +  7.00 y_{11} & -6.00 y_{12} & +  6.00 y_{13} & -5.00 y_{14} & -10.00 y_{15}\\
 y_{4}   &  1.0 & -4.00 y_{9} & +  3.00 y_{10} & -9.00 y_{11} & +  2.00 y_{12} & -10.00 y_{13} &   & -3.00 y_{15}\\
 y_{5}   &  1.0 & -6.00 y_{9} & -7.00 y_{10} & -7.00 y_{11} & +  6.00 y_{12} & -5.00 y_{13} & -7.00 y_{14} & +  7.00 y_{15}\\
 y_{6}   &  1.0 & +  2.00 y_{9} & -3.00 y_{10} & -7.00 y_{11} & -3.00 y_{12} & +  1.00 y_{13} & +  3.00 y_{14} & +  9.00 y_{15}\\
 y_{7}   &  1.0 & -6.00 y_{9} & -9.00 y_{10} & -2.00 y_{11} & -5.00 y_{12} & +  7.00 y_{13} & +  4.00 y_{14} & -1.00 y_{15}\\
 y_{8}   &  1.0 & -8.00 y_{9} & -10.00 y_{10} & +  4.00 y_{11} & -9.00 y_{12} & +  5.00 y_{13} & +  1.00 y_{14} & +  2.00 y_{15}\\
\hline
z    &  -0 & + 15.00 y_{9} & +  3.00 y_{10} & -12.00 y_{11} & + 36.00 y_{12} & -64.00 y_{13} & -7.00 y_{14} & +  4.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 5 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.44186046512 & -1.47 y_{8} & -7.47 y_{10} & +  1.02 y_{2} & -4.56 y_{9} & +  5.09 y_{13} & +  4.56 y_{14} & +  8.98 y_{11}\\
 y_{15}   &  0.0232558139535 & +  0.19 y_{8} & +  0.19 y_{10} & -0.21 y_{2} & +  0.02 y_{9} & +  1.16 y_{13} & -1.02 y_{14} & -1.79 y_{11}\\
 y_{3}   &  0.0697674418605 & -1.44 y_{8} & + 13.56 y_{10} & +  2.37 y_{2} & +  8.07 y_{9} & -10.51 y_{13} & +  5.93 y_{14} & + 24.63 y_{11}\\
 y_{4}   &  1.16279069767 & -0.70 y_{8} & +  0.30 y_{10} & +  0.53 y_{2} & -5.84 y_{9} & -11.86 y_{13} & +  2.84 y_{14} & -3.53 y_{11}\\
 y_{5}   &  1.86046511628 & +  0.88 y_{8} & -12.12 y_{10} & -1.74 y_{2} & -11.14 y_{9} & +  8.02 y_{13} & -14.86 y_{14} & -19.26 y_{11}\\
 y_{6}   &  0.860465116279 & +  1.88 y_{8} & +  1.88 y_{10} & -1.74 y_{2} & +  4.86 y_{9} & +  9.02 y_{13} & -5.86 y_{14} & -23.26 y_{11}\\
 y_{7}   &  0.395348837209 & +  0.16 y_{8} & -3.84 y_{10} & +  0.44 y_{2} & -1.60 y_{9} & +  1.77 y_{13} & +  5.60 y_{14} & -0.44 y_{11}\\
 y_{12}   &  0.116279069767 & -0.07 y_{8} & -1.07 y_{10} & -0.05 y_{2} & -0.88 y_{9} & +  0.81 y_{13} & -0.12 y_{14} & +  0.05 y_{11}\\
\hline
z    &  4.27906976744 & -1.77 y_{8} & -34.77 y_{10} & -2.51 y_{2} & -16.72 y_{9} & -30.05 y_{13} & -15.28 y_{14} & -17.49 y_{11}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.76744186047 & +  1.47 x_{1} & -0.19 x_{15} & +  1.44 x_{3} & +  0.70 x_{4} & -0.88 x_{5} & -1.88 x_{6} & -0.16 x_{7} & +  0.07 x_{12}\\
 x_{10}   &  34.7674418605 & +  7.47 x_{1} & -0.19 x_{15} & -13.56 x_{3} & -0.30 x_{4} & + 12.12 x_{5} & -1.88 x_{6} & +  3.84 x_{7} & +  1.07 x_{12}\\
 x_{2}   &  2.51162790698 & -1.02 x_{1} & +  0.21 x_{15} & -2.37 x_{3} & -0.53 x_{4} & +  1.74 x_{5} & +  1.74 x_{6} & -0.44 x_{7} & +  0.05 x_{12}\\
 x_{9}   &  16.7209302326 & +  4.56 x_{1} & -0.02 x_{15} & -8.07 x_{3} & +  5.84 x_{4} & + 11.14 x_{5} & -4.86 x_{6} & +  1.60 x_{7} & +  0.88 x_{12}\\
 x_{13}   &  30.0465116279 & -5.09 x_{1} & -1.16 x_{15} & + 10.51 x_{3} & + 11.86 x_{4} & -8.02 x_{5} & -9.02 x_{6} & -1.77 x_{7} & -0.81 x_{12}\\
 x_{14}   &  15.2790697674 & -4.56 x_{1} & +  1.02 x_{15} & -5.93 x_{3} & -2.84 x_{4} & + 14.86 x_{5} & +  5.86 x_{6} & -5.60 x_{7} & +  0.12 x_{12}\\
 x_{11}   &  17.488372093 & -8.98 x_{1} & +  1.79 x_{15} & -24.63 x_{3} & +  3.53 x_{4} & + 19.26 x_{5} & + 23.26 x_{6} & +  0.44 x_{7} & -0.05 x_{12}\\
\hline
z    &  -4.27906976744 & -1.44 x_{1} & -0.02 x_{15} & -0.07 x_{3} & -1.16 x_{4} & -1.86 x_{5} & -0.86 x_{6} & -0.40 x_{7} & -0.12 x_{12}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.76744186047 & +  1.47 x_{1} & -0.19 x_{15} & +  1.44 x_{3} & +  0.70 x_{4} & -0.88 x_{5} & -1.88 x_{6} & -0.16 x_{7} & +  0.07 x_{12}\\
 x_{10}   &  34.7674418605 & +  7.47 x_{1} & -0.19 x_{15} & -13.56 x_{3} & -0.30 x_{4} & + 12.12 x_{5} & -1.88 x_{6} & +  3.84 x_{7} & +  1.07 x_{12}\\
 x_{2}   &  2.51162790698 & -1.02 x_{1} & +  0.21 x_{15} & -2.37 x_{3} & -0.53 x_{4} & +  1.74 x_{5} & +  1.74 x_{6} & -0.44 x_{7} & +  0.05 x_{12}\\
 x_{9}   &  16.7209302326 & +  4.56 x_{1} & -0.02 x_{15} & -8.07 x_{3} & +  5.84 x_{4} & + 11.14 x_{5} & -4.86 x_{6} & +  1.60 x_{7} & +  0.88 x_{12}\\
 x_{13}   &  30.0465116279 & -5.09 x_{1} & -1.16 x_{15} & + 10.51 x_{3} & + 11.86 x_{4} & -8.02 x_{5} & -9.02 x_{6} & -1.77 x_{7} & -0.81 x_{12}\\
 x_{14}   &  15.2790697674 & -4.56 x_{1} & +  1.02 x_{15} & -5.93 x_{3} & -2.84 x_{4} & + 14.86 x_{5} & +  5.86 x_{6} & -5.60 x_{7} & +  0.12 x_{12}\\
 x_{11}   &  17.488372093 & -8.98 x_{1} & +  1.79 x_{15} & -24.63 x_{3} & +  3.53 x_{4} & + 19.26 x_{5} & + 23.26 x_{6} & +  0.44 x_{7} & -0.05 x_{12}\\
\hline
z    &  -0.279069767442 & -4.44 x_{1} & +  0.98 x_{15} & -6.07 x_{3} & -2.16 x_{4} & +  4.14 x_{5} & + 13.14 x_{6} & -3.40 x_{7} & -0.12 x_{12}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  1.76744186047 & +  1.47 x_{1} & -0.19 x_{15} & +  1.44 x_{3} & +  0.70 x_{4} & -0.88 x_{5} & -1.88 x_{6} & -0.16 x_{7} & +  0.07 x_{12}\\
 x_{10}   &  34.7674418605 & +  7.47 x_{1} & -0.19 x_{15} & -13.56 x_{3} & -0.30 x_{4} & + 12.12 x_{5} & -1.88 x_{6} & +  3.84 x_{7} & +  1.07 x_{12}\\
 x_{2}   &  2.51162790698 & -1.02 x_{1} & +  0.21 x_{15} & -2.37 x_{3} & -0.53 x_{4} & +  1.74 x_{5} & +  1.74 x_{6} & -0.44 x_{7} & +  0.05 x_{12}\\
 x_{9}   &  16.7209302326 & +  4.56 x_{1} & -0.02 x_{15} & -8.07 x_{3} & +  5.84 x_{4} & + 11.14 x_{5} & -4.86 x_{6} & +  1.60 x_{7} & +  0.88 x_{12}\\
 x_{13}   &  30.0465116279 & -5.09 x_{1} & -1.16 x_{15} & + 10.51 x_{3} & + 11.86 x_{4} & -8.02 x_{5} & -9.02 x_{6} & -1.77 x_{7} & -0.81 x_{12}\\
 x_{14}   &  15.2790697674 & -4.56 x_{1} & +  1.02 x_{15} & -5.93 x_{3} & -2.84 x_{4} & + 14.86 x_{5} & +  5.86 x_{6} & -5.60 x_{7} & +  0.12 x_{12}\\
 x_{11}   &  17.488372093 & -8.98 x_{1} & +  1.79 x_{15} & -24.63 x_{3} & +  3.53 x_{4} & + 19.26 x_{5} & + 23.26 x_{6} & +  0.44 x_{7} & -0.05 x_{12}\\
\hline
z    &  -0.279069767442 & -4.44 x_{1} & +  0.98 x_{15} & -6.07 x_{3} & -2.16 x_{4} & +  4.14 x_{5} & + 13.14 x_{6} & -3.40 x_{7} & -0.12 x_{12}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{8} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  2.0 & +  1.66 x_{1} & -0.21 x_{15} & +  1.63 x_{3} & +  0.79 x_{4} & -1.13 x_{8} & -2.13 x_{6} & -0.18 x_{7} & +  0.08 x_{12}\\
 x_{10}   &  59.0 & + 27.55 x_{1} & -2.74 x_{15} & +  6.21 x_{3} & +  9.26 x_{4} & -13.71 x_{8} & -27.71 x_{6} & +  1.61 x_{7} & +  2.03 x_{12}\\
 x_{2}   &  6.0 & +  1.87 x_{1} & -0.16 x_{15} & +  0.47 x_{3} & +  0.84 x_{4} & -1.97 x_{8} & -1.97 x_{6} & -0.76 x_{7} & +  0.18 x_{12}\\
 x_{9}   &  39.0 & + 23.03 x_{1} & -2.37 x_{15} & + 10.11 x_{3} & + 14.63 x_{4} & -12.61 x_{8} & -28.61 x_{6} & -0.45 x_{7} & +  1.76 x_{12}\\
 x_{13}   &  14.0 & -18.39 x_{1} & +  0.53 x_{15} & -2.58 x_{3} & +  5.53 x_{4} & +  9.08 x_{8} & +  8.08 x_{6} & -0.29 x_{7} & -1.45 x_{12}\\
 x_{14}   &  45.0 & + 20.08 x_{1} & -2.11 x_{15} & + 18.32 x_{3} & +  8.89 x_{4} & -16.82 x_{8} & -25.82 x_{6} & -8.34 x_{7} & +  1.29 x_{12}\\
 x_{11}   &  56.0 & + 22.95 x_{1} & -2.26 x_{15} & +  6.79 x_{3} & + 18.74 x_{4} & -21.79 x_{8} & -17.79 x_{6} & -3.11 x_{7} & +  1.47 x_{12}\\
\hline
z    &  8.0 & +  2.42 x_{1} & +  0.11 x_{15} & +  0.68 x_{3} & +  1.11 x_{4} & -4.68 x_{8} & +  4.32 x_{6} & -4.16 x_{7} & +  0.21 x_{12}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{13} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  3.26180257511 & -0.09 x_{13} & -0.16 x_{15} & +  1.40 x_{3} & +  1.29 x_{4} & -0.31 x_{8} & -1.40 x_{6} & -0.21 x_{7} & -0.05 x_{12}\\
 x_{10}   &  79.9699570815 & -1.50 x_{13} & -1.95 x_{15} & +  2.35 x_{3} & + 17.54 x_{4} & -0.11 x_{8} & -15.61 x_{6} & +  1.17 x_{7} & -0.14 x_{12}\\
 x_{2}   &  7.42203147353 & -0.10 x_{13} & -0.10 x_{15} & +  0.21 x_{3} & +  1.40 x_{4} & -1.05 x_{8} & -1.15 x_{6} & -0.79 x_{7} & +  0.04 x_{12}\\
 x_{9}   &  56.5250357654 & -1.25 x_{13} & -1.71 x_{15} & +  6.88 x_{3} & + 21.55 x_{4} & -1.24 x_{8} & -18.49 x_{6} & -0.81 x_{7} & -0.05 x_{12}\\
 x_{1}   &  0.761087267525 & -0.05 x_{13} & +  0.03 x_{15} & -0.14 x_{3} & +  0.30 x_{4} & +  0.49 x_{8} & +  0.44 x_{6} & -0.02 x_{7} & -0.08 x_{12}\\
 x_{14}   &  60.2818311874 & -1.09 x_{13} & -1.53 x_{15} & + 15.50 x_{3} & + 14.93 x_{4} & -6.91 x_{8} & -17.00 x_{6} & -8.66 x_{7} & -0.29 x_{12}\\
 x_{11}   &  73.4649499285 & -1.25 x_{13} & -1.61 x_{15} & +  3.57 x_{3} & + 25.63 x_{4} & -10.46 x_{8} & -7.71 x_{6} & -3.47 x_{7} & -0.33 x_{12}\\
\hline
z    &  9.8426323319 & -0.13 x_{13} & +  0.17 x_{15} & +  0.34 x_{3} & +  1.83 x_{4} & -3.49 x_{8} & +  5.38 x_{6} & -4.20 x_{7} & +  0.02 x_{12}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{5}   &  10.8571428571 & -0.63 x_{13} & +  0.12 x_{15} & -9.98 x_{1} & +  4.29 x_{4} & +  4.61 x_{8} & +  2.98 x_{6} & -0.37 x_{7} & -0.84 x_{12}\\
 x_{10}   &  92.7142857143 & -2.41 x_{13} & -1.47 x_{15} & -16.74 x_{1} & + 22.57 x_{4} & +  8.15 x_{8} & -8.26 x_{6} & +  0.91 x_{7} & -1.46 x_{12}\\
 x_{2}   &  8.57142857143 & -0.18 x_{13} & -0.06 x_{15} & -1.51 x_{1} & +  1.86 x_{4} & -0.31 x_{8} & -0.49 x_{6} & -0.82 x_{7} & -0.08 x_{12}\\
 x_{9}   &  93.8571428571 & -3.92 x_{13} & -0.31 x_{15} & -49.05 x_{1} & + 36.29 x_{4} & + 22.97 x_{8} & +  3.05 x_{6} & -1.58 x_{7} & -3.91 x_{12}\\
 x_{3}   &  5.42857142857 & -0.39 x_{13} & +  0.20 x_{15} & -7.13 x_{1} & +  2.14 x_{4} & +  3.52 x_{8} & +  3.13 x_{6} & -0.11 x_{7} & -0.56 x_{12}\\
 x_{14}   &  144.428571429 & -7.10 x_{13} & +  1.63 x_{15} & -110.56 x_{1} & + 48.14 x_{4} & + 47.66 x_{8} & + 31.56 x_{6} & -10.40 x_{7} & -8.99 x_{12}\\
 x_{11}   &  92.8571428571 & -2.63 x_{13} & -0.88 x_{15} & -25.48 x_{1} & + 33.29 x_{4} & +  2.11 x_{8} & +  3.48 x_{6} & -3.87 x_{7} & -2.34 x_{12}\\
\hline
z    &  11.7142857143 & -0.27 x_{13} & +  0.24 x_{15} & -2.46 x_{1} & +  2.57 x_{4} & -2.28 x_{8} & +  6.46 x_{6} & -4.23 x_{7} & -0.17 x_{12}\\
\end{array}\]


 $ x_{4} $ enters and Unbounded Dictionary!


 $ x_{4} $ enters and Unbounded Dictionary!
\end{document}
